Lami's theorem

In physics, Lami's theorem is an equation relating the magnitudes of three coplanar, concurrent and non-collinear vectors, which keeps an object in static equilibrium, with the angles directly opposite to the corresponding vectors. According to the theorem,

${\frac {v_{A}}{\sin \alpha }}={\frac {v_{B}}{\sin \beta }}={\frac {v_{C}}{\sin \gamma }}$

where $v_{A},v_{B},v_{C}$ are the magnitudes of the three coplanar, concurrent and non-collinear vectors, ${\vec {v}}_{A},{\vec {v}}_{B},{\vec {v}}_{C}$, which keep the object in static equilibrium, and $\alpha ,\beta ,\gamma$ are the angles directly opposite to the vectors, thus satisfying $\alpha +\beta +\gamma =360^{o}$.

Lami's theorem is applied in static analysis of mechanical and structural systems. The theorem is named after Bernard Lamy.

Proof

As the vectors must balance ${\vec {v}}_{A}+{\vec {v}}_{B}+{\vec {v}}_{C}={\vec {0}}$, hence by making all the vectors touch its tip and tail the result is a triangle with sides $v_{A},v_{B},v_{C}$ and angles $180^{o}-\alpha ,180^{o}-\beta ,180^{o}-\gamma$ ($\alpha ,\beta ,\gamma$ are the exterior angles).

By the law of sines then

${\frac {v_{A}}{\sin(180^{o}-\alpha )}}={\frac {v_{B}}{\sin(180^{o}-\beta )}}={\frac {v_{C}}{\sin(180^{o}-\gamma )}}.$

Then by applying that for any angle $\theta$, $\sin(180^{o}-\theta )=\sin \theta$ (supplementary angles have the same sine), and the result is

${\frac {v_{A}}{\sin \alpha }}={\frac {v_{B}}{\sin \beta }}={\frac {v_{C}}{\sin \gamma }}.$

See also

• Mechanical equilibrium
• Parallelogram of force
• Tutte embedding

Further reading

• R.K. Bansal (2005). "A Textbook of Engineering Mechanics". Laxmi Publications. p. 4. ISBN 978-81-7008-305-4.
• I.S. Gujral (2008). "Engineering Mechanics". Firewall Media. p. 10. ISBN 978-81-318-0295-3